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Try startingStarting with $dG=-SdT+VdP$ So, we have $$S=-\left(\frac{\partial G}{\partial T}\right)$$$$\left(\frac{\partial G}{\partial T}\right)_P=-S\tag{1}$$ SoBut, from the equation $$\left(\frac{\partial S}{\partial T}\right)_P=-\left(\frac{\partial^2 G}{\partial T^2}\right)_P$$$G = H - TS$, we also have:$$\left(\frac{\partial G}{\partial T}\right)_P=\left(\frac{\partial H}{\partial T}\right)_P-S-T\left(\frac{\partial S}{\partial T}\right)_P\tag{2}$$ If we combine Eqns. 1 and 2, we obtain:$$\left(\frac{\partial H}{\partial T}\right)_P=T\left(\frac{\partial S}{\partial T}\right)_P\tag{3}$$The left hand side of Eqn. 3 is the definition of $C_P$. So,$$C_P=T\left(\frac{\partial S}{\partial T}\right)_P\tag{4}$$ QED

Try starting with $dG=-SdT+VdP$ So, $$S=-\left(\frac{\partial G}{\partial T}\right)$$ So, $$\left(\frac{\partial S}{\partial T}\right)_P=-\left(\frac{\partial^2 G}{\partial T^2}\right)_P$$

Starting with $dG=-SdT+VdP$, we have $$\left(\frac{\partial G}{\partial T}\right)_P=-S\tag{1}$$ But, from the equation $G = H - TS$, we also have:$$\left(\frac{\partial G}{\partial T}\right)_P=\left(\frac{\partial H}{\partial T}\right)_P-S-T\left(\frac{\partial S}{\partial T}\right)_P\tag{2}$$ If we combine Eqns. 1 and 2, we obtain:$$\left(\frac{\partial H}{\partial T}\right)_P=T\left(\frac{\partial S}{\partial T}\right)_P\tag{3}$$The left hand side of Eqn. 3 is the definition of $C_P$. So,$$C_P=T\left(\frac{\partial S}{\partial T}\right)_P\tag{4}$$ QED

    Post Undeleted by Chet Miller
    Post Deleted by Chet Miller
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Try starting with $dG=-SdT+VdP$ So, $$S=-\left(\frac{\partial G}{\partial T}\right)$$ So, $$\left(\frac{\partial S}{\partial T}\right)_P=-\left(\frac{\partial^2 G}{\partial T^2}\right)_P$$

Try starting with $dG=-SdT+VdP$

Try starting with $dG=-SdT+VdP$ So, $$S=-\left(\frac{\partial G}{\partial T}\right)$$ So, $$\left(\frac{\partial S}{\partial T}\right)_P=-\left(\frac{\partial^2 G}{\partial T^2}\right)_P$$

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Try starting with $dG=-SdT+VdP$